It depends on the emoji service you got.
Emoji Art: All the rights belong to you. You can do anything and there is no need to give us a credit.
Emoji App (Stickers & Keyboard). This is different because the emojis will be in an app development. DoggyMojis will have the exclusive rights of your pet image only related to the app development, that’s all. The final art rights of the emojis are owned by DoggyMojis who can’t use the emojis for any other purpose outside the app.
You have the right to use the emojis and artwork only for free purpose but you must give the credit to DoggyMojis. Any Images used by you for any commercial purpose not directly related to the app or of those terms - as merchandising - must be with the express permission (a text confirmation will be enough) of DoggyMojis and may include the payment of additional fees as royalties.